Instabilities in Bounded and Composite Polytropes

Unbounded, Complete Polytropes

Free-Energy Function and Its Derivatives

The free-energy function that is relevant to a discussion of the structure and stability of unbounded configurations having polytropic index, <math>~n</math>, has the form,

<math>~\mathcal{G}(x)</math>

<math> -ax^{-1} +b x^{-3/n} + \mathcal{G}_0 \, , </math>

where <math>~x \equiv R/R_\mathrm{SWS}</math> identifies the radius of the configuration and <math>\mathcal{G}_0</math> is an arbitrary constant. If the coefficients, <math>~a, b</math>, and <math>~c</math>, are held constant while varying the configuration's size, we see that,

<math>~\frac{d\mathcal{G}}{dx}</math>

<math> x^{-2} \biggl[ a - \frac{3b}{n}\cdot x^{(n-3)/n} \biggr] \, , </math>

and,

<math>~\frac{d^2\mathcal{G}}{dx^2}</math>

<math> x^{-3} \biggl[ -2a + \frac{3(3+n)b}{n^2}\cdot x^{(n-3)/n} \biggr] \, . </math>

In terms of the system's mass and its structural form factors, <math>~\mathfrak{f}_M</math>, <math>~\mathfrak{f}_A</math>, and <math>~\mathfrak{f}_W</math>, the two relevant coefficients are,

<math>~a</math>	<math>~=</math>	<math>~\frac{3}{5}\biggl(\frac{n+1}{n}\biggr) \biggl[ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr) \cdot \frac{1}{\mathfrak{f}_M} \biggr]^2 \mathfrak{f}_W \, ,</math>
<math>~b</math>	<math>~=</math>	<math>~n \biggl( \frac{3}{4\pi}\biggr)^{1/n} \biggl[ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr) \cdot \frac{1}{\mathfrak{f}_M} \biggr]^{(n+1)/n} \mathfrak{f}_A \, . </math>

Equilibrium Radius

A configuration's equilibrium radius, <math>~x_\mathrm{eq}</math>, can be determined one of two ways:

Extrema in the Free Energy

Equilibria are identified by extrema in the free-energy function. Setting <math>d\mathcal{G}/dx = 0</math>, we find,

<math>~x_\mathrm{eq}</math>

<math> \biggl(\frac{an}{3b} \biggr)^{n/(n-3)} \, , </math>

Related Discussions

Wikipedia introduction to the Lane-Emden equation
Wikipedia introduction to Polytropes

User:Tohline/SSC/Stability BoundedCompositePolytropes

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